An organic compound $\left({\mathrm{C}}_8{\mathrm{H}}_{10}{\mathrm{O}}_2\right)$ rotates plane-polarized light. It produces pink color with neutral ${\mathrm{FeCl}}_3$ solution. What is the total number of all the possible isomers for this compound?

In the reaction,
\(
2 \mathrm{X}+\mathrm{B}_2 \mathrm{H}_6 \longrightarrow\left[\mathrm{BH}_2(\mathrm{X})_2\right]^{+}\left[\mathrm{BH}_4\right]^{-}
\)
The amine (s) \(X\), is (are)

The number of stereoisomers obtained by bromination of trans-2-butene is

The compound (s) having peroxide linkage is(are) :

For a double strand DNA, one strand is given below:

The amount of energy required to split the double strand DNA into two single strands is _______ kcal \(\mathrm{mol}^{-1}.\)
[Given: Average energy per \(\mathrm{H}\)-bond for A-T base pair \(=1.0 \mathrm{kcal} \mathrm{mol}^{-1}, \mathrm{G}-\mathrm{C}\) base pair \(=1.5 \mathrm{kcal}\) \(\mathrm{mol}^{-1}\), and A-U base pair \(=1.25 \mathrm{kcal} \mathrm{mol}^{-1}.\) Ignore electrostatic repulsion between the phosphate groups.]

Which of the following liberates ${\mathrm{O}}_2$ upon hydrolysis?

According to Bohr's model, the highest kinetic energy is associated with the electron in the

A trinitro compound, $1,3,5$-tris-($4$-nitrophenyl) benzene, on complete reaction with an excess of $\mathrm{Sn}/\mathrm{HCl}$ gives a major product, which on treatment with an excess of ${\mathrm{NaNO}}_2/\mathrm{HCl}$ at $0°\mathrm{C}$ provides $\mathbf{P}$ as the product. $\mathbf{P}$, upon treatment with excess of ${\mathrm{H}}_2\mathrm{O}$ at room temperature, gives the product $\mathbf{Q}$. Bromination of $\mathbf{Q}$ in aqueous medium furnishes the product $\mathbf{R}$. The compound $\mathbf{P}$ upon treatment with an excess of phenol under basic conditions gives the product $\mathbf{S}$. The molar mass difference between compounds $\mathbf{Q}$ and $\mathbf{R}$ is $474 \mathrm{g} {\mathrm{mol}}^{-1}$ and between compounds $\mathbf{P}$ and $\mathbf{S}$ is $172.5 \mathrm{g} {\mathrm{mol}}^{-1}$.
The number of heteroatoms present in one molecule of \(R\) is [Use: Molar mass (in \(\text{g mol} -1\) ): \(H =1,\) \(C =12, N=14, O =16,\) \(Br =80, Cl =35.5\) Atoms other than C and H are considered as heteroatoms]

Which of the following halides react(s) with \(\mathrm{AgNO}_{3}(\mathrm{aq})\) to give a precipitate that dissolves in \(\mathrm{Na}_{2} \mathrm{~S}_{2} \mathrm{O}_{3}(\mathrm{aq})\) ?

Match the Statements/Expressions in Column I with the Statements/Expressions in Column II.

Column I	Column II
(A) The minimum value of \(\frac{x^2+2 x+4}{x+2}\) is	(p) 0
(B) Let \(A\) and \(B\) be \(3 \times 3\) matrices of real numbers, where \(A\) is symmetric, \(B\) is skew-symmetric and \((A+B)(A-B) =\) \((A-B)(A+B)\). If \((A B)^t=(-1)^k\) \(A B\), where \((A B)^t\) is the transpose of the matrix \(A B\), then possible values of \(k\) are	(q) 1
(C) Let \(a=\log _3 \log _3 2\). An integer \(k\) satisfying \(1<2^{\left(-k+3^{-a}\right)}<2\), must be less than	(r) 2
(D) If \(\sin \theta=\cos \phi\), then the possible values of \(\frac{1}{\pi}\left(\theta \pm \phi-\frac{\pi}{2}\right)\) are	(s) 3

Paragraph:
Consider the functions defined implicitly by the equation \(y^3-3 y+x=0\) on various intervals in the real line. If \(x \in(-\infty,-2) \cup(2, \infty)\), the equation implicitly defines a unique real valued differentiable function \(y=f(x)\). If \(x \in(-2,2)\), the equation implicitly defines a unique real valued differentiable function \(y=g(x)\), satisfying \(g(0)=0\).Question:
If \(f(-10 \sqrt{2})=2 \sqrt{2}\), then \(f^{\prime \prime}(-10 \sqrt{2})\) is equal to

A point source \(\mathrm{S}\) emits unpolarized light uniformly in all directions. At two points \(\mathrm{A}\) and \(\mathrm{B}\), the ratio \(r=I_A / I_B\) of the intensities of light is 2. If a set of two polaroids having \(45^{\circ}\) angle between their pass-axes is placed just before point \(\mathrm{B}\), then the new value of \(r\) will be ______

For any $3\times 3$ matrix $M$, let $\left|M\right|$ denote the determinant of $M$. Let $I$ be the $3\times 3$ identity matrix. Let $E$ and $F$ be two $3\times 3$ matrices such that $(I-EF)$ is invertible. If $G=(I-EF{)}^{-1}$, then which of the following statements is (are) TRUE ?